Unscented propagationUnscented propagationof measurement uncertainty

Jan Peter Hessling , Jörgen Stenarson, Thomas Svensson

1

Thomas SvenssonSP Technical Research Institute of Sweden,

Sweden

Of current interest…Non-linear propagation of uncertainty

• Willink & Hall, Metrologia

22 qqq +=• Poor result – poor assumptions/methods!

22

21 qqq +=

• Not a mathematical mystery• …but has mathematical complexity

Application Model equation

Microwave reflection

Common models strongly non-linear-in- parameters

<<+= qqh 122

21sqrt

Logarithmic perception:Audio, micro-wave, tele-communication etc.

Saddle point

Fatigue life time

( )[ ]

( )

=>>=

+=

=

<<

==

<+⋅=

−

−

−

N FF

N

qFFhh

qqh

qh

2

18

,

1,10

100,log20

000life

21sdl

2

3

dB

β

αεαε

Dynamic measurement ( )∏∏=

=

kk

kk

sb

saqsH ,

1.0β

General set up for the illustrated examples

• Symmetric confidence intervals of scalar measurements are unique!• Such intervals can be estimated with various methods.• Symmetric confidence interval:

[ ], +− hhhh

• NOT the common(?) definition of symmetric confidence interval:

• All parameters uniformly distributed:

[ ]( ) ( ) ( ) ( )1,0,21ProbProb

,

∈−=+>=−<

+−

PPhhhhhh

hhhh

PP

PP

( ) ( )[ ]( ) ( ) ( )1,0,1ProbProb

,

∈−=+>+−<

+−

PPhhhhhh

hqhhqh

PP

PP

• All parameters uniformly distributed:

( )

>≤≤−

=1,0

11,21

q

qqfq

!!!

( ) )( qhqhh ≠≠

Strong bias of the GUM confidence interval !

Measurand Center symm. conf . int .

Center of rangeof meas.

Mean of mapping

Mappingof mean

Symmetricconfidence interval

conf . int . of meas.

( ) ( ) [ ]

]13.41.3[9.31.58.74.710

]0.700.70-[0000

]60-89-[100688075

]1.260.18[076.071.072.0

,

30life

sdl

dB

sqrt

⋅

−−−−

+−

hh

h

h

h

hhhhqhqhhhh PPC

AmbitionGUM

Ambitionof GUM?

Finite rangerequired…Desired! Focus here!

’Arbitrary’ bias of the GUM confidence interval !

• Strong dependence on parameterization!

( ) ( ) '''' qhqhh ( ) ( )( )( )( )

( )( ) 76.012.1

76.076.0

76.082.0

76.00,

''''

05.01022

22

21sqrt

22

21sqrt

22

2121sqrt

wqqwh

vqqvh

uqquh

qqqqh

qhqhh

=+≡=+≡

=+≡+=

( )( ) 76.012.105.01022

21sqrt wqqwh =+≡

GUMAmbitionof GUM?

Poor width of the GUM confidence interval !

• The GUM relies strongly upon regularity of the measurementequations and fast convergence of Taylor expansions!

Measurand Half -width of symm . Halfwidth GUMMeasurand Half -width of symm . conf. int.

Halfwidth GUM

080.0

15

54.0

2

sdl

dB

sqrt

−−

≈

h

h

h

cckhh qPP σσ

1.W

hy n

o re

sult?

2. H

ighe

ror

der?

8.10.610

080.03

0life

sdl

⋅hh

h

Desired!

GUM unable to distinguish hfrom ’0’ in almost all respects !!!

Differences in general interest

• Traditional statistics: Moments (mean, variance etc.)Primary tools: Statistics, detailed pdf’sSecondary tools: Multi-variable calculus, linear algebra

• Metrology: Confidence interval of measurementPrimary tools: Multi-variable calculus, linear algebra Secondary tools: Statistics

The ’physics ’ of confidence intervals differs from The ’physics ’ of confidence intervals differs from the ’physics’ of moments!

Not utilized at all in the GUM!

RESTART…- ILLUSTRATE (1)- ILLUSTRATE (1)AND SOLVE (2)

THE MYSTERIOUS ’DONUT’:

( ) 22, qqqqh +=( ) 22

2121sqrt , qqqqh +=

Level surfaceRange of parameters

1.5

( ) 22

2121sqrt , qqqqh +=

0.5

1

1.5

h

sqrth

Phh +sqrt

-1-0.5

00.5

1

-1

0

1

0

q1

q2

Phh −sqrt

( )( )2,1maxmin, q

Project level surface onto parameter space…

1.5

0.5

1

1.5

h

-1-0.5

00.5

1

-1

0

1

0

q1

q2

Confidence DomainConfidence Boundary

1

( ) 22

2121sqrt , qqqqh +=

2 CONFIDENCE BOUNDARIES

-0.5

0

0.5q

2

CONFIDENCE DOMAIN’DONUT’

NOT REALIZABLE!

-1 -0.5 0 0.5 1

-1

-0.5

q1

Confidence DomainConfidence Boundary

1

( ) 22

2121sqrt , qqqqh +=

2 CONFIDENCE BOUNDARIES

SAMPLE !=> CONFIDENCE INTERVAL

OF MEASUREMENT !

-0.5

0

0.5q

2

CONFIDENCE DOMAIN’DONUT’

P=0.95NOT

REALIZABLE!

-1 -0.5 0 0.5 1

-1

-0.5

q1

P=0.95

Problem described, Problem described, but not yet solved!

Sampling of pdf…

• Random sampling – Monte Carlo – you know!• One class of alternatives of deterministic sampling

1) Sample multi-variate pdf with a finite number of points such that:

1

kσthat:

2) Propagate samples3) Statistical analysis:

• Another class of alternatives of deterministic sampling1) Estimate confidence domain (parameter space)

( ) ∑∫=

==N

k

mkkq

mm WN

dqqfqq1

1 σ

( )kh σ ( )( )( )

−=

=

σσ

σ

σσ

22 hhh

hh

1) Estimate confidence domain (parameter space)2) Sample the estimated confidence domain3) Propagate samples4) Correct/adjust with statistical analysis:

kλ

The sets of samples forms an incomplete, but useful(!) representation of the pdf of uncertain parameters

( )kk λσ or

( )( )( )

−=

=

λ

λ

λ

λ2

hhh

hh

P

( )kh λ

Sampling of pdf…

• Random sampling – Monte Carlo – you know!• One class of alternatives of deterministic sampling

1) Sample multi-variate pdf with a finite number of points such that:

1

kσUNSCENTED KALMAN FILTER:that:

2) Propagate samples3) Statistical analysis:

• Another class of alternatives of deterministic sampling1) Estimate confidence domain (parameter space)

( ) ∑∫=

==N

k

mkkq

mm WN

dqqfqq1

1 σ

( )kh σ ( )( )( )

−=

=

σσ

σ

σσ

22 hhh

hh

UNSCENTED KALMAN FILTER:PROPAGATION OF COVARIANCE

(JULIER, UHLMANN 1995)

UNSCENTED PROPAGATION OF 1) Estimate confidence domain (parameter space)2) Sample the estimated confidence domain3) Propagate samples4) Correct/adjust with statistical analysis:

kλ

The sets of samples forms an incomplete, but useful(!) representation of the pdf of uncertain parameters

( )kk λσ or

( )( )( )

−=

=

λ

λ

λ

λ2

hhh

hh

P

( )kh λ

UNSCENTED PROPAGATION OF MEASUREMENT UNCERTAINTY

(Hessling, Svensson, Stenarsson)

Sampling of pdf, approximation??? 1(2)PDF:• Propagation of pd-functions is difficult• GUM does not propagate pdfs in any other ways than by assumptions• GUM has an ’unscented appearance’ (propagates ’points’)!• Unscented approaches utilize incomplete representations of pdfs

Reflect available knowledge !

MODEL:• Standard GUM makes bold assumptions of models.

Indisputable only for linear propagation!• Unscented propagation makes no approximation of the model• Unscented propagation makes no approximation of the model

Does not rely upon regularity in any way!

Sampling of pdf, approximation??? 2(2)

PERFORMANCE / SIMPLICITY• The ambition of unscented propagation provides a [much] more

effective and transferable(!) alternative, than MC• Simplest unscented methods:• Simplest unscented methods:

– Are significantly simpler than the GUM– Do not rely upon nearly as many assumptions

• Accuracy as well as simplicity may be the reason to ’go unscented’• Conventional wisdom:

Anyone without education in statistics would ’go unscent ed’!

GUM approximates some unscented alternatives, generally not the reverse!

Another example: Result

Performance:26% bias

Efficiency:10 000 000 realizations

( ) 321, qqyqyh ∝

20

Performance:0.2% biasEfficiency:

2realizations

Into details: Monotonicity / Connectivity

-65

-60

-55( )[ ]αε +⋅= qh log20dB

-85

-80

-75

-70

-65

h dB• Model equation not monotonic

• Confidence domain disconnected !

-1 -0.5 0 0.5 1-100

-95

-90

q

Require monotonicity !

How to:1) Split parameter space into ’monotonic subdomains’!2) Parameterize (mirror, rotate) consistently for mapping2) Parameterize (mirror, rotate) consistently for mapping3) For each subdomain:

a) Select points to propagate(’sample pdf’/’estimate conf. domain’)

b) Map all pointsc) Statistical analysis of mapped points

4) Weight results according to accumulated probability4) Weight results according to accumulated probability

Into details: Monotonicity / Connectivity

-60

-55

( )[ ]αε +⋅= qh log20dB

( )( ) ±=≈⋅+ ∑ shWhsh skkP ,

2,λ

-95

-90

-85

-80

-75

-70

-65

h dB

( ) ±=≈⋅+ ∑=

shWhshk

kP ,1

λ

Estimatedconfidenceinterval

Correctconfidenceinterval

[ ] [ ]]60-89-[]60-89-[

,, PPPP hhhhhhhh +−+−

-1 -0.5 0 0.5 1-100

q

( )−,1λ ( )+,1λ( )+,2λ ( )−,2λ

]60-89-[]60-89-[Exact!

1D: Exact(!) unscented propagation throughmonotonic measurement equations• One uncertain

model parameter ( )qfqLambda-

points

• Uncertainmeasurement ( )hfhmonotonic

h

1.

2.

24

[ ]PP hhhh +− ,

CORRECT !

Monotonicity for arbitrary number of dimensions?

• Parameterize all possible testlines through midpoint, over the range• Monotonicity well defined along each line• Generalized monotonic (GM) = monotonic along all such lines• GM NOT equivalent to connected confidence domains!• GM NOT equivalent to connected confidence domains!

(GM stronger requirement)

0.5

1 ’Donut’

is not [ generalized]

( ) 22

2121sqrt , qqqqh +=

-1 -0.5 0 0.5 1

-1

-0.5

0

q1

q2 is not [ generalized]

monotonic!- Even though the confidence domainis connected!

Divide intomonotonic subdomains ( ) 2

22121sqrt , qqqqh +=

1

-0.5

0

0.5q

212

3 4

Split donut is[generalized] monotonic!…and symmetric– solve one quadrant!

-1 -0.5 0 0.5 1

-1

-0.5

q1

3 4

Estimate confidence domain for quadrant 1…sample it… and propagate samples…

• Estimate confidence region with a linear approximation of h.• Do not approximate h!!!• Sample the estimated confidence region (lambda-points)

⇒ Non-degenerate unscented propagation

( )

( ) ( ) ( )( )( )[ ] ( )( ) ( )( )[ ]s

s

s

sPP

qq

T

qPqP

s

yhyhhhhh

hhqhkhk

ssq

λλσδ

δλ

,max,,min,

~~cov

~~,

=+−∇∇∇=∇=

±=⋅+=

∇

Normalized gradient

⇒ Non-degenerate unscented propagationof measurement uncertainty

(for linear approx. of confidence domain)

Estimates of symmetric confidence intervalResult for the donut problem

GUM Non-degenerate Symmetric

( ) 22

2121sqrt , qqqqh +=

GUM Non-degenerateunscentedpropagation

Symmetricconfidence interval

No split

Split intomonotonic

[ ] [ ] [ ]

]1.260.18[]1.260.16[]0.890.11[

N/AN/A]FailFail[

,,, PPPPPP hhhhhhhhhhhh +−+−+−

monotonic ]1.260.18[]1.260.16[]0.890.11[Acceptable

Relation to GUM

• Linearize measurement equation to propagate lambda-points:

• Then,

( ) ( ) ( ) hqqhh T ∇⋅−+= λλ

( ) ( ) ( )• Then,

• …which is the GUM expression!

( ) ( ) ( )( ) ( )( )hqhkh

qhhhhT

PP ∇∇⋅=

===

covλλ

λλ

29

GUM => Non-Deg. Unscented prop…

30

GUM <= Non-Deg. Unscented prop…

• Confidence interval:

( )( ) ( ) ( )( ) ( ) ( )sTs

s

s

qkh

qyhqyhyhh

ΕΕ⋅=

≠≠=

cov

,,,,

λ

• Signed unscented sensitivities:

( ) ( ) ( )s

sTsPP qkh ΕΕ⋅= cov,

( ) ( )( )[ ] ( )[ ] 1

cov,−

−⋅=Ε TP

Tss UqUkUhyhs λ

Increment vector in original basisVariation of meas.

Related but different results for non-linear measurements!

31

Difference quotient

Relation to Monte Carlo

• UKF bears ’superficial resemblance’ to Monte Carlo simulations,

• Differences:− Sigma-points are not drawn at random, − Sigma-points are not drawn at random,

but with a very specific algorithm− The weights are controlled independently of pdf

• Consequences: − ’Convergence superior’, compared to random

sampling (MC)− Transferability simplifies ’enormously’

Unscented propagation is a sparsedeterministic variant of MC!

32

1.5

Estimated confidence domain

-0.5

0

0.5

1q

2

λλλλλλλλ

λλλλλλλλ

λλλλ λλλλ

λλλλλλλλ

qc

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

q1

λλλλλλλλ

1.5

Estimated confidence domain

-0.5

0

0.5

1q

2

λλλλλλλλ

λλλλλλλλ

λλλλ λλλλ

λλλλλλλλ

qc

P=0.95

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

q1

λλλλλλλλP=0.95

Summary• Topic:Non-linear propagation of model uncertainty• Fact: The GUM approach may give exceedingly poor confidence

intervals• For cases of large practical interest!• May fail completely!

• Proposed solution: Non-degenerate unscented propagation• Straight-forward generalization to any model equation• Source of estimation error easily illustrated• Performance

• Accurate• Simple (degenerate) / comparable to GUM (non-deg.)• Simple (degenerate) / comparable to GUM (non-deg.)• Few assumptions of pdf• No assumption of model equation• Harmonized to common available knowledge

35Thank you!